How to generate regular expression for language of regular expressions?

Question

Alphabets = $\{0, 1, \epsilon, \emptyset, [, ], \cup, \circ, \star\}$ where $\circ$ stands for concatenation, $\star$ is kleene star, $\epsilon$ is empty string, $[$ and $]$ are brackets to be used for language to be generated. Only those Regular Expressions to be considered which don't have nested brackets $[[]]$.

How to handle $[ \ ]$ in this problem ? An expression can be - $[1\star\star \cup 0] \cup [\epsilon \cup 1 \circ 0] \cup 1\star$

What would be a regular expression that describes the language of such regular expressions?

Edit -

I want to come up with Regular Expression for the Language of Regular Expressions as show in example above. One thing though is that the regular expression should not generate such expressions which contains nested bracket ( we don't want to consider that ) so $[1 \cup [0 \cup \epsilon]]$ is not valid and should not be generated by the Regular Expression. What I have done and my understanding so far - For single character Regular Expression would be $(0 \cup 1 \cup \epsilon \cup \phi)$. Be reminded that we are creating regular expression for Language of Regular Expressions. So $(0 \cup 1 \cup \epsilon \cup \phi)$ accepts these regular expressions - $0$ , $1$ , $\epsilon$ , $\phi$.

For $0\star\star$ kind of expressions - $(0 \cup 1 \cup \epsilon \cup \phi)\star^{\star}$ should work.

For $0\star\star \cup 1\star$ types we can have $(0 \cup 1 \cup \epsilon \cup \phi)\star^{\star} \bigcup\ (0 \cup 1 \cup \epsilon \cup \phi)\star^{\star}$

I am getting feeling that somewhere I need to use recursion. So I how to handle bracket $[ ]$ in this case ? How to accept those expressions which have brackets and they are not nested ? If anyone can give me logic or direction to solve this problem competely that will also be ok.

Answer 1

Let us consider first a simpler situation, in which you have only one "atom" $a$, only one binary operator $\circ$, and no unary operator. The expressions you are captured by $$ X (\circ X)^*, \quad X = a+[a(\circ a)^*]. $$ In order to get an actual regular expression, substitute $X$ (surrounded by parentheses) in the left expression.

Adding a postfix operation $\star$ isn't too difficult: $$ Y (\circ Y)^*, \quad Y = A+[A(\circ A)^*]\star^*, \quad A= a\star^*. $$

In order to get an actual regular expression, first substitute $A$ into $Y$, then $Y$ into the leftmost expression.

Finally, the expression you are looking for only differs in that there are several "atoms" and several binary operations: $$ Z (\circ Z)^*, \quad Z = B+[B((\cup + \circ) B)^*]\star^*, \quad B= (0+1+\underline{\epsilon}+\underline{\emptyset})\star^*. $$ Here $\underline{\epsilon},\underline{\emptyset}$ are the versions in your alphabet.